You can follow this tutorial here: http://www.raywenderlich.com/1626/how-to-post-to-a-users-wall-upload-photos-and-add-a-like-button-from-your-iphone-app
It shows you everything you need to do to set up Like box with your iOS app.
Question
Is there a way to create a like button inside a native iOS app for Facebook page?
Solution
You can follow this tutorial here: http://www.raywenderlich.com/1626/how-to-post-to-a-users-wall-upload-photos-and-add-a-like-button-from-your-iphone-app
It shows you everything you need to do to set up Like box with your iOS app.
OTHER TIPS
See the url:https://developers.facebook.com/docs/ios/like-button
This is the easiest way to show the like button in ios app but it still beta feature.
Here is the code:
[FBSettings enableBetaFeature:FBBetaFeaturesLikeButton];
[FBSettings enablePlatformCompatibility:NO];
FBLikeControl *like = [[FBLikeControl alloc] init];
like.objectID = @"http://shareitexampleapp.parseapp.com/photo1/";
like.likeControlHorizontalAlignment=FBLikeControlHorizontalAlignmentRight;
like.likeControlStyle=FBLikeControlStyleBoxCount;
[self.view addSubview like];
This is no longer possible since 2017... could be possible another way.
reference: https://marketingland.com/facebook-will-disable-like-button-third-party-mobile-apps-228159